# Anatole Katok

Penn State University

October 20, 2006 from 16:00 to 18:00 (Montreal/Miami time) On location

Colloquium presented by **Anatole Katok (Penn State University)**

In the classical theory of dynamical systems which deals with diffeomorphisms and smooth flows on compact manifolds hyperbolic behavior is known to imply stability of the global topological orbit structure under small perturbations of the system, called structural stability. However, differentiable orbit structure is never stable. Furthermore, even topological stability has to be qualified in the continuous time case where time change must be allowed. And full hyperbolic structure is necessary for structural stability. It is quite remarkable that for actions of higher rank abelian groups much stronger rigidity phenomena appear. First, there is global rigidity of differentiable orbit structure for standard examples of actions with global hyperbolic behavior, such as commuting hyperbolic automorphisms of a torus, or Weyl chamber flows (which are higher rank counterparts of geodesic flows on symmetric spaces of negative curvature). Notice that for R^k actions with k>1 only linear time changes are allowed. Rigidity for these actions was established in the mid-1990ies in a series of papers joint with M. Guysinsky and R. Spatzier. The central idea of the method is proving regularity of the structural stability maps by building invariant geometric structures for perturbed actions and showing that the topological conjugacy must intertwine the invariant structures for perturbed and unperturbed actions. More recently jointly with D. Damjanovic we showed that similar differentiable rigidity takes place for several classes of partially hyperbolic actions where there is no structural stability in the rank one case and hence the previous method is totally useless. I Instead we developed two mutually complementary methods which may even be more interesting that the results they produce. One method is based on linearization of the conjugacy equation, solving the linearized problem with tame estimates (based on vanishing of the obstructions due to higher rank) and using KAM (Kolmogorov-Arnold-Moser) type iteration scheme to construct a converging sequence of approximate conjugacies. The other method is based on translating the conjugacy problem to a cohomology problem over the perturbed action and using description of generators and relations in classical split Lie groups to solve those equations.

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